Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 22 June 2019 doi:10.20944/preprints201906.0225.v1 Article On 2-Inner Product Spaces and Reproducing Kernel Property Saeed Hashemi Sababe 1 1 Young Researchers and Elite Club, Malard Branch, Islamic Azad University, Malard, Iran.; Hashemi [email protected] Received: date; Accepted: date; Published: date Abstract: This paper is devoted to the study of reproducing kernels on 2-inner product Hilbert spaces. We focus on a new structure to produce reproducing kernel Hilbert and Banach spaces. According to multi-variable computing, this structures can be useful in electrocardiographs, machine learning and economy. Keywords: 2-inner product; vector-valued spaces; 2-semi norm MSC: Primary 47B32; Secondary 47A70 1. Introduction Reproducing Kernel Hilbert Spaces (RKHS) have been found unbelievably useful in several branch of abstract and objective sciences like the machine learning community. This theory developed for quite some time and has been used in especially in statistics for about twenty years. More recently, their application extended in to objective theories and arithmetics algorithms, deeply. distribution theory is the most relation between abstract concepts and objective applications. (See [7,10]). A reproducing kernel Hilbert space (RKHS in short) is a Hilbert space H of functions defined on a fixed set X such that for each x 2 X the evaluation functional at x, i.e., Ex( f ) := f (x), f 2 H, is continuous on H. The Riesz representation theorem gives a unique function K : X × X ! C such that • fk(., x) : x 2 Xg ⊂ H, • f (x) = h f , k(., x)iH, x 2 X, f 2 H. The function K is called the reproducing kernel of H. Similar structure can be defined for Banach spaces with a quick difference. More details about these kind of spaces can be found in [1,8,12]. In the following, in section 2, we prepare some preliminaries of 2-inner product spaces and review some important and useful theorems and lemmas. In section 3, we defined 2-inner product reproducing kernel Hilbert spaces and proved some theorems. finally, we extended the theorems to 2-inner reproducing Banach spaces. 2. Preliminaries In this section, we review some basic concepts and definitions. we also, recall some important lemmas and theorems. Among these content, Riesz representation theorem on 2-inner product spaces plays a key roll in the next sections. This theorem, is a wise extension of basic type in [9] to 2-inner product spaces. Let V be a linear space of dimension greater than 1 over the field F (either R or C). The function h., .; .i : V × V × V ! F is called a 2-inner product if the following conditions holds, 1. hx, x; zi ≥ 0 and hx, x; zi = 0 iff x and z are linearly dependent. 2. hx, x; zi = hz, z; xi. 3. hy, x; zi = hx, y; zi. 4. hax, y; zi = ahx, y; zi, for all scalars a 2 F. h i h i h i 5. x1 + x2, y; z = x1, y; z + x2, y; z . © 2019 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 22 June 2019 doi:10.20944/preprints201906.0225.v1 2 of 8 Therefore, the pair (V, h., .; .i) is called a 2-inner product space. Let (V, h., .; .i) be a 2-inner product space and x, y, b 2 X then x ?b y , hx, y; bi = 0. We define a 2-norm on V × V by kx, yk2 = hx, x; yi. It is easy to see that, this norm satisfies the following conditions: 1. kx, zk ≥ 0 and kx, zk = 0 if and only if x and z are linearly dependent. 2. kx, zk = kz, xk, 3. kax, zk = jajkx, zk for all a 2 F, k k ≤ k k k k 4. x1 + x2, z x1, z + x2, z . Therefore the pair (V, k., .k) is called a linear 2-normed space. 2-norms have several interesting properties. More details can be found in [5]. If (V, h., .i) is an inner product space, then the standard 2-inner product h., .; .i is defined on V by hx, yi hx, zi hx, y; zi = = hx, yihz, zi − hx, zihz, yi. hz, yi hz, zi h i f g ≤ ≤ Let (V, ., .; . ) be a 2-inner product space over F. If ei , 1 i n are linearly independent vectors f g ≤ ≤ 2 in the 2-inner product space V, then ei , 1 i n is called a b-orthonormal set if for b V, h i 6 h i ≤ ≤ h i ei, ej; b = 0 for i = j and ei, ej; b = 1 for i = j where 1 i n. Let (V, ., .; . ) be a 2-inner product space over F. The subspace of all linearly independent vectors in the 2-inner product space V respect to x 2 V is closed. Let (H, h., .; .i) be a 2-inner product space over F, b 2 H, then 1. A sequence fxng in H is said to be a b-Cauchy sequence if for every e > 0 there exists N > 0 such that for every m, n ≥ N, 0 < kxn − xm, bk < e . 2. H is said to be b-Hilbert if every b-Cauchy sequence is convergent in the semi-normed space (H, k., bk). Let (V, k., .k) be a 2-normed space. Let W be a subspace of V, b 2 V be fixed, then a map T : W × hbi ! F is called a b-linear functional on W × hbi whenever for every x, y 2 W and a 2 F holds 1. T(x + y, b) = T(x, b) + T(y, b), 2. T(ax, b) = aT(x, b). A b-linear functional T : W × hbi ! F is said to be bounded if there exists a real number M > 0 such that jT(x, b)j ≤ Mkx, bk for every x 2 W. The norm of the b-linear functional T : W × hbi ! F is defined by kTk = in f fM > 0; jT(x, b)j ≤ Mjx, bj, 8x 2 Wg. It can be seen that, kTk = supfjT(x, b)j; kx, bk ≤ 1g, kTk = supfjT(x, b)j; kx, bk = 1g, kTk = supfjT(x, b)j; kx, bk 6= 0g. and jT(x, b)j ≤ kTkkx, bk. k k 6 2 ∗ For a 2-normed space (V, ., . ) and 0 = b V, Vb denote the Banach space of all bounded b-linear functionals on V × hbi, where hbi is the subspace of V generated by b. Let V be a vector space over Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 22 June 2019 doi:10.20944/preprints201906.0225.v1 3 of 8 2 2 − 2 h i F. Let b V and y1, y2 V, then y1 is said to be b-congruent to y2 iff (y1 y2) b , the subspace generated by b. (Riesz Representation Theorem on 2-Inner Product Spaces) Let H be a b-Hilbert 2 ∗ 2 h i space and T Hb then there exists a unique y H up to b-congruence such that T(x, b) = x, y; b and kTk = ky, bk. 3. Main Theorems In this section, we try to extend the concept of reproducing kernel to 2-inner product spaces. This extension is along with the previews concepts. The main goal of this extension is to construct a new space with more applicable properties. Let X be a set and H be a linear space of two variable functions f : X × X × X ! R endowed with a 2-inner product. Fix g 2 H. So we can speak about g-Hilbert space Hg. Let Hg be a subspace of Hg such that all 3-dimensional evaluation E(x,y)( f ) = f (x, y), be bounded. So by theorem (2), there exist a function kx,y such that h i E(x,y)( f ) = f (x, y) = f , k(x,y); g . Now we can define a kernel function K : X × X × X ! R by h i K(x, y, z) = k(x,y), k(y,z); g . In this case h i k k2 K(x, x, x) = k(x,x), k(x,x); g = k(x,x), g . The pair (Hg, K) is called a 2-inner product reproducing kernel Hilbert space (2IPRKHS in short). 2 An examples of 2-inner product reproducing kernel Hilbert space is as follows: Let H2 be the two variable Hardy-Hilbert space, consists of all analytic functions having power series representations H 2 with square-summable complex coefficients and I be a subspace of H2 such that ( ) ¥ ¥ H n n j j2 ¥ I = f : f (z1, z2) = ∑ anz1 z2 : a1 = 0 and ∑ an < , n=0 n=0 ∑¥ n n ∑¥ n n Where I denotes the identity function, f (z1, z2) = n=0 anz1 z2 and h(z1, z2) = n=0 bnz1 z2 . The H 2-inner product on I is defined by ¥ h i f , h; I = a0b0 + ∑ anbn. n=2 ¥ k (t t ) = + ∑ znzntntn In this case, let (z1,z2) 1, 2 1 n=2 1 2 1 2 , then f (z z ) = h f k Ii 1, 2 , (z1,z2); , and ¥ K(z z z ) = hk k Ii = + ∑ jz j2nznzn 1, 2, 3 (z1,z2), (z2,z3); 1 2 1 3 . n=2 We follow by explaining some properties of 2-inner product reproducing kernels. If Hg is a 2IPRKHS on X with reproducing kernel K(x, y, z), then K(x, y, x) = K(y, x, y).